Quantifying aggregation of IgE-FcepsilonRI by multivalent antigen

A general model of multivalent binding with ligands of heterotypic subunits and multiple surface receptors

Multivalent cell surface receptor binding is a ubiquitous biological phenomenon with functional and therapeutic significance. Predicting the amount of ligand binding for a cell remains an important question in computational biology as it can provide great insight into cell-to-cell communication and rational drug design toward specific targets. In this study, we extend a mechanistic, two-step multivalent binding model. This model predicts the behavior of a mixture of different multivalent ligand complexes binding to cells expressing various types of receptors. It accounts for the combinatorially large number of interactions between multiple ligands and receptors, optionally allowing a mixture of complexes with different valencies and complexes that contain heterogeneous ligand units. We derive the macroscopic predictions and demonstrate how this model enables large-scale predictions on mixture binding and the binding space of a ligand. This model thus provides an elegant and computationally efficient framework for analyzing multivalent binding.

A spatio-temporal model reveals self-limiting FcɛRI cross-linking by multivalent antigens

Aggregation of cell surface receptor proteins by multivalent antigens is an essential early step for immune cell signalling. A number of experimental and modelling studies in the past have investigated multivalent ligand-mediated aggregation of IgE receptors (FcɛRI) in the plasma membrane of mast cells. However, understanding of the mechanisms of FcɛRI aggregation remains incomplete. Experimental reports indicate that FcɛRI forms relatively small and finite-sized clusters when stimulated by a multivalent ligand. By contrast, modelling studies have shown that receptor cross-linking by a trivalent ligand may lead to the formation of large receptor superaggregates that may potentially give rise to hyperactive cellular responses. In this work, we have developed a Brownian dynamics-based spatio-temporal model to analyse FcɛRI aggregation by a trivalent antigen. Unlike the existing models, which implemented non-spatial simulation approaches, our model explicitly accounts for the coarse-grained site-specific features of the multivalent species (molecules and complexes). The model incorporates membrane diffusion, steric collisions and sub-nanometre-scale site-specific interaction of the time-evolving species of arbitrary structures. Using the model, we investigated temporal evolution of the species and their diffusivities. Consistent with a recent experimental report, our model predicted sharp decay in species mobility in the plasma membrane in response receptor cross-linking by a multivalent antigen. We show that, due to such decay in the species mobility, post-stimulation receptor aggregation may become self-limiting. Our analysis reveals a potential regulatory mechanism suppressing hyperactivation of immune cells in response to multivalent antigens.

Dissecting FcγR Regulation through a Multivalent Binding Model

Many immune receptors transduce activation across the plasma membrane through their clustering. With Fcγ receptors (FcγRs), this clustering is driven by binding to antibodies of differing affinities that are in turn bound to multivalent antigen. As a consequence of this activation mechanism, accounting for and rationally manipulating immunoglobulin (Ig)G effector function is complicated by, among other factors, differing affinities between FcγR species and changes in the valency of antigen binding. In this study, we show that a model of multivalent receptor-ligand binding can effectively account for the contribution of IgG-FcγR affinity and immune complex valency. This model in turn enables us to make specific predictions about the effect of immune complexes of defined composition. In total, these results enable both rational immune complex design for a desired IgG effector function and the deconvolution of effector function by immune complexes.

Mathematical Modeling of Bioassays

The high affinity and specificity of biological receptors determine the demand for and the intensive development of analytical systems based on use of these receptors. Therefore, theoretical concepts of the mechanisms of these systems, quantitative parameters of their reactions, and relationships between their characteristics and ligand-receptor interactions have become extremely important. Many mathematical models describing different bioassay formats have been proposed. However, there is almost no information on the comparative characteristics of these models, their assumptions, and predictive insights. In this review we suggested a set of criteria to classify various bioassays and reviewed classical and contemporary publications on these bioassays with special emphasis on immunochemical analysis systems as the most common and in-demand techniques. The possibilities of analytical and numerical modeling are discussed, as well as estimations of the minimum concentrations that may be detected in bioassays and recommendations for the choice of assay conditions.

Air Pollution and Climate Change Effects on Allergies in the Anthropocene: Abundance, Interaction, and Modification of Allergens and Adjuvants

Air pollution and climate change are potential drivers for the increasing burden of allergic diseases. The molecular mechanisms by which air pollutants and climate parameters may influence allergic diseases, however, are complex and elusive. This article provides an overview of physical, chemical and biological interactions between air pollution, climate change, allergens, adjuvants and the immune system, addressing how these interactions may promote the development of allergies. We reviewed and synthesized key findings from atmospheric, climate, and biomedical research. The current state of knowledge, open questions, and future research perspectives are outlined and discussed. The Anthropocene, as the present era of globally pervasive anthropogenic influence on planet Earth and, thus, on the human environment, is characterized by a strong increase of carbon dioxide, ozone, nitrogen oxides, and combustion- or traffic-related particulate matter in the atmosphere. These environmental factors can enhance the abundance and induce chemical modifications of allergens, increase oxidative stress in the human body, and skew the immune system toward allergic reactions. In particular, air pollutants can act as adjuvants and alter the immunogenicity of allergenic proteins, while climate change affects the atmospheric abundance and human exposure to bioaerosols and aeroallergens. To fully understand and effectively mitigate the adverse effects of air pollution and climate change on allergic diseases, several challenges remain to be resolved. Among these are the identification and quantification of immunochemical reaction pathways involving allergens and adjuvants under relevant environmental and physiological conditions.

IgE immunotherapy: a novel concept with promise for the treatment of cancer

The importance of antibodies in activating immune responses against tumors is now better appreciated with the emergence of checkpoint blockade antibodies and with engineered antibody Fc domains featuring enhanced capacity to focus potent effector cells against cancer cells. Antibodies designed with Fc regions of the IgE class can confer natural, potent, long-lived immune surveillance in tissues through tenacious engagement of high-affinity cognate Fc receptors on distinct, often tumor-resident immune effector cells, and through ability to activate these cells under tumor-induced Th2-biased conditions. Here, we review the properties that make IgE a contributor to the allergic response and a critical player in the protection against parasites, which also support IgE as a novel anti-cancer modality. We discuss IgE-based active and passive immunotherapeutic approaches in disparate in vitro and in vivo model systems, collectively suggesting the potential of IgE immunotherapies in oncology. Translation toward clinical application is now in progress.

Optimal aggregation of FcεRI with a structurally defined trivalent ligand overrides negative regulation driven by phosphatases

To investigate why responses of mast cells to antigen-induced IgE receptor (FcεRI) aggregation depend nonlinearly on antigen dose, we characterized a new artificial ligand, DF3, through complementary modeling and experimentation. This ligand is a stable trimer of peptides derived from bacteriophage T4 fibritin, each conjugated to a hapten (DNP). We found low and high doses of DF3 at which degranulation of mast cells sensitized with DNP-specific IgE is minimal, but ligand-induced receptor aggregation is comparable to aggregation at an intermediate dose, optimal for degranulation. This finding makes DF3 an ideal reagent for studying the balance of negative and positive signaling in the FcεRI pathway. We find that the lipid phosphatase SHIP and the protein tyrosine phosphatase SHP-1 negatively regulate mast cell degranulation over all doses considered. In contrast, SHP-2 promotes degranulation. With high DF3 doses, relatively rapid recruitment of SHIP to the plasma membrane may explain the reduced degranulation response. Our results demonstrate that optimal secretory responses of mast cells depend on the formation of receptor aggregates that promote sufficient positive signaling by Syk to override phosphatase-mediated negative regulatory signals.

Recombinant IgE antibodies for passive immunotherapy of solid tumours: from concept towards clinical application

Therapeutic antibodies have revolutionised treatment of some cancers and improved prognosis for many patients. Over half of those available are approved for haematological malignancies, but efficacious antibodies for solid tumours are still urgently needed. Clinically available antibodies belong to the IgG class, the most prevalent antibody class in human blood, while other classes have not been extensively considered. We hypothesised that the unique properties of IgE, a class of tissue-resident antibodies commonly associated with allergies, which can trigger powerful immune responses through strong affinity for their particular receptors on effector cells, could be employed for passive immunotherapy of solid tumours such as ovarian and breast carcinomas. Our laboratory has examined this concept by evaluating two chimaeric antibodies of the same specificity (MOv18) but different isotype, an IgG1 and an IgE against the tumour antigen folate receptor α (FRα). The latter demonstrates the potency of IgE to mount superior immune responses against tumours in disease-relevant models. We identified Fcε receptor-expressing cells, monocytes/macrophages and eosinophils, activated by MOv18 IgE to kill tumour cells by mechanisms such as ADCC and ADCP. We also applied this notion to a marketed therapeutic, the humanised IgG1 antibody trastuzumab and engineered an IgE counterpart, which retained the functions of trastuzumab in restricting proliferation of HER2/neu-expressing tumour cells but also activated effector cells to kill tumour cells by different mechanisms. On-going efficacy, safety evaluations and future first-in-man clinical studies of IgE therapeutics constitute key metrics for this concept, providing new scope for antibody immunotherapies for solid tumours.

Modeling multivalent ligand-receptor interactions with steric constraints on configurations of cell-surface receptor aggregates

We use flow cytometry to characterize equilibrium binding of a fluorophore-labeled trivalent model antigen to bivalent IgE-FcepsilonRI complexes on RBL cells. We find that flow cytometric measurements are consistent with an equilibrium model for ligand-receptor binding in which binding sites are assumed to be equivalent and ligand-induced receptor aggregates are assumed to be acyclic. However, this model predicts extensive receptor aggregation at antigen concentrations that yield strong cellular secretory responses, which is inconsistent with the expectation that large receptor aggregates should inhibit such responses. To investigate possible explanations for this discrepancy, we evaluate four rule-based models for interaction of a trivalent ligand with a bivalent cell-surface receptor that relax simplifying assumptions of the equilibrium model. These models are simulated using a rule-based kinetic Monte Carlo approach to investigate the kinetics of ligand-induced receptor aggregation and to study how the kinetics and equilibria of ligand-receptor interaction are affected by steric constraints on receptor aggregate configurations and by the formation of cyclic receptor aggregates. The results suggest that formation of linear chains of cyclic receptor dimers may be important for generating secretory signals. Steric effects that limit receptor aggregation and transient formation of small receptor aggregates may also be important.

A non-chromatographic method for the purification of a bivalently active monoclonal IgG antibody from biological fluids

This paper describes a method for the purification of monoclonal antibodies (rat anti-2,4-dinitrophenyl IgG: IgG(DNP); and mouse antidigoxin IgG: IgG(Dgn)) from ascites fluid. This procedure (for IgG(DNP)) has three steps: (i) precipitation of proteins heavier than immunoglobulins with ammonium sulfate; (ii) formation of cyclic complexes of IgG(DNP) by causing it to bind to synthetic multivalent haptens containing multiple DNP groups; (iii) selective precipitation of these dimers, trimers, and higher oligomers of the target antibody, followed by regeneration of the free antibody. This procedure separates the targeted antibody from a mixture of antibodies, as well as from other proteins and globulins in a biological fluid. This method is applicable to antibodies with a wide range of monovalent binding constants (0.1 microM to 0.1 nM). The multivalent ligands we used (derivatives of DNP and digoxin) isolated IgG(DNP) and IgG(Dgn) from ascites fluid in yields of >80% and with >95% purity. This technique has two advantages over conventional chromatographic methods for purifying antibodies: (i) it is selective for antibodies with two active Fab binding sites (both sites are required to form the cyclic complexes) over antibodies with one or zero active Fab binding sites; (ii) it does not require chromatographic separation. It has the disadvantage that the structure of the hapten must be compatible with the synthesis of bi- and/or trivalent analogues.

Trivalent antigens for degranulation of mast cells

Degranulation of basophils and mast cells plays a central role in allergic reactions. Degranulation is a response to cell surface receptor aggregation caused by association of receptors with antibodies bound to multivalent antigens. Tools used in studying this process have included small-molecule divalent antigens, but they suffer from weak signaling apparently due to small aggregate size. We have prepared trivalent antigens that allow formation of larger aggregates and potent responses from mast cells.

Cell surface dynamics: the balance between diffusion, aggregation and endocytosis

The cell membrane lies at the interface between an extracellular set of signals and the appropriate intracellular response. Specifically, lymphocyte activity is determined by the spatial and structural response to antigens, as mediated by cell surface receptors. In order to correlate experimentally observed cellular activities, such as secretion, anergy, death, survival and division to external stimuli, it is necessary to monitor cell surface dynamics. B-lymphocyte activation results from the stimulation by large immune complexes comprising antigens, B-cell receptors (BcRs) and co-receptors. Compartmentalisation of the interacting molecular components is required in order to assure the rapid initiation of specialised and sustained signalling cascades. In this study, a Monte Carlo simulation of the cell membrane dynamics was developed to clarify the receptor dynamics, aggregation mechanisms and their combined effect on cellular functions. This simulation is based on experimentally measured parameters and represents a feasible, advanced and reliable framework to investigate the cell surface. The current study focussed on B-cell surface dynamics. A model demonstrating the basic properties of BcR dynamics and how BcR kinetics is affected by lipid rafts is developed. The authors studied BcR interactions with multivalent ligands and the influence of lipid rafts on this interaction. Finally, the dynamics of the initial steps of BcR-mediated cell activation is estimated and the effect of the association of signalling molecules with lipid rafts is demonstrated. These results are used to suggest some novel hypotheses on BcR-mediated B-cell activation.

Small heat shock proteins differentially affect Abeta aggregation and toxicity

beta-Amyloid (Abeta) is the primary protein component of senile plaques in Alzheimer's disease (AD) and has been implicated in neurotoxicity associated with the disease. Abeta aggregates readily in vitro and in vivo, and its toxicity has been linked to its aggregation state. Prevention of Abeta aggregation has been investigated as a means to prevent Abeta toxicity associated with AD. Recently we found that Hsp20 from Babesia bovis prevented both Abeta aggregation and toxicity [S. Lee, K. Carson, A. Rice-Ficht, T. Good, Hsp20, a novel alpha-crystallin, prevents Abeta fibril formation and toxicity, Protein Sci. 14 (2005) 593-601.]. In this work, we examined the mechanism of Hsp20 interaction with Abeta1-40 and compared its activity to that of other small heat shock proteins, carrot Hsp17.7 and human Hsp27. While all three small heat shock proteins were able to prevent Abeta aggregation, only Hsp20 was able to attenuate Abeta toxicity in cultured SH-SY5Y cells. Understanding the mechanism of the Hsp20-Abeta interaction may provide insights into the design of the next generation of Abeta aggregation and toxicity inhibitors.

Binding of multivalent CD147 phage induces apoptosis of U937 cells

CD147 is a broadly expressed cell-surface molecule and serves as a signaling receptor for extracellular cyclophilins. CD147 also appears to interact with immune cells, but its counter-receptor on these cells has not been clearly described. In the present report, we displayed multiple copies of the CD147 extracellular domain (CD147Ex) on VCSM13 phage to study the interaction of CD147 with its ligand. Recognition of phage containing fusion protein of CD147Ex and gpVIII (CD147Ex phage) by four different anti-CD147 mAbs indicated that at least parts of the CD147 are properly folded. Specific binding of CD147Ex phage to various cell types was demonstrated by flow cytometry. Morphological changes, however, were observed only in U937, a monocytic cell line, after 24 h incubation with multivalent CD147Ex phage. After 48 h, U937 cell propagation ceased. Staining with annexin V and the presence of cleaved caspase-3 indicated that many of the CD147Ex phage-treated cells had lost viability through apoptotic cell death. The above results suggest that CD147 induces apoptosis in U973 cells and that at least a portion of this cell death program involves a caspase-dependent pathway.

A model of neutralization of Chlamydia trachomatis based on antibody and host cell aggregation on the elementary body surface

Humoral immunity is that aspect of specific immunity that is mediated by B lymphocytes and involves the neutralizing of pathogens by means of antibodies attaching to the pathogen's binding sites. Antibodies bind to and block ligand sites on the pathogen which prevents these sites from attaching to target cell receptors and so cell entry is inhibited. Many studies investigate the role of humoral immunity for protection against chlamydial challenge and they have shown that neutralization of the chlamydial body requires a large number of attached antibodies. Steric hindrance greatly influences the number of available sites that may be bound, reducing relative occupancy well below 100%. We model steric effects of antibody Fab fragment attachment indicating that they must be taken into consideration to accurately model valency, the number of available binding sites. We derive a partial differential equation for the number of antibody Fabs and host cell receptors that are aggregated to extracellular chlamydial elementary bodies. We consider steric effects in describing the size distribution of aggregates. Our theory is in good agreement with Monte Carlo simulations of binding. We use our theoretical prediction for the valency in a model for the in-host population dynamics of a chlamydial infection and we fit our model to experimental data.

Investigation of early events in Fc epsilon RI-mediated signaling using a detailed mathematical model

Aggregation of Fc epsilon RI on mast cells and basophils leads to autophosphorylation and increased activity of the cytosolic protein tyrosine kinase Syk. We investigated the roles of the Src kinase Lyn, the immunoreceptor tyrosine-based activation motifs (ITAMs) on the beta and gamma subunits of Fc epsilon RI, and Syk itself in the activation of Syk. Our approach was to build a detailed mathematical model of reactions involving Fc epsilon RI, Lyn, Syk, and a bivalent ligand that aggregates Fc(epsilon)RI. We applied the model to experiments in which covalently cross-linked IgE dimers stimulate rat basophilic leukemia cells. The model makes it possible to test the consistency of mechanistic assumptions with data that alone provide limited mechanistic insight. For example, the model helps sort out mechanisms that jointly control dephosphorylation of receptor subunits. In addition, interpreted in the context of the model, experimentally observed differences between the beta- and gamma-chains with respect to levels of phosphorylation and rates of dephosphorylation indicate that most cellular Syk, but only a small fraction of Lyn, is available to interact with receptors. We also show that although the beta ITAM acts to amplify signaling in experimental systems where its role has been investigated, there are conditions under which the beta ITAM will act as an inhibitor.

Biosensor immunosurface engineering inspired by B-cell membrane-bound antibodies: modeling and analysis of multivalent antigen capture by immobilized antibodies

Immobilized antibodies are used by many biosensors and diagnostic tests as specific receptors for the presence of targeted substances in clinical, biological, or environmental samples. The antibodies used in these devices are the soluble form of the antibodies presented on the B-cell membrane: they have the same specificity, but they may differ from those presented on the B cell in orientation, flexibility, mobility, and support-membrane properties. These properties influence the formation of noncovalent bonds between the pathogen antigenic determinants (epitopes) and the amino acids of the antibodies. This paper extends the theoretical modeling foundation addressing multivalent antigen binding to cell surface receptors to account for local and far-field antibody surface density effects, immobilized antibodies, and the flexibility and range of motion of immobilized antibodies. An analysis of the derived model provides insight into the design of biosensor immunosurfaces to enhance pathogen capture capability.

A quantitative approach for studying IgE-FcepsilonRI aggregation

Aggregation of cell surface receptors is a ubiquitous means of initiating signal transduction in many cellular systems. In this manuscript, we describe a combined theoretical and experimental approach based on multiparameter flow cytometry for measuring the time course of ligand induced aggregation of IgE-FcepsilonRI on RBL cells. By fluorescently labeling both the ligand and surface IgE (sIgE), we have developed an assay that permits us to simultaneously measure both occupancy of sIgE combining sites and association of antigen with the cell surface. This allows for a direct calculation of the degree of receptor aggregation present on the cell. By employing new mixing technologies developed for flow cytometry, we are able to look at aggregation in the sub second time domain. To extend our work, we have synthesized a new set of chemically well defined ligands (of valences 1-3) to use as probes in our studies. We show that the magnitude of the cellular response is dramatically increased as the valence of our ligand is raised from two to three.

Flow cytometric analysis of ligand-receptor interactions and molecular assemblies

Flow cytometers make homogeneous real-time measurements of ligand-receptor interactions and, simultaneously, the physiological responses of cells. Their multiparameter capabilities are also useful in resolving multicomponent assemblies or in developing multiplexed assays. Recent advances suggest that these approaches can be extended in several important ways. Sample delivery in the millisecond time domain is applicable to the analysis of complex binding kinetics and reaction mechanisms. The homogeneous discrimination of free components and particle-based assemblies can be extended into the micromolar concentration range. Measurements can be made of molecular assemblies among proteins, DNA, RNA, lipids, and carbohydrates on beads. The topography and assembly of components within cells can be evaluated with resonance energy transfer. Temperature dependence can be evaluated with Peltier temperature control. Many assembly endpoints can be assessed through new tools for high-throughput flow cytometry using plate-based assay formats and small volume samples.

Retention of antigen on follicular dendritic cells and B lymphocytes through complement-mediated multivalent ligand-receptor interactions: theory and application to HIV treatment

In HIV-infected patients, large quantities of HIV are associated with follicular dendritic cells (FDCs) in lymphoid tissue. During antiretroviral therapy, most of this virus disappears after six months of treatment, suggesting that FDC-associated virus has little influence on the eventual outcome of long-term therapy. However, a recent theoretical study using a stochastic model for the interaction of HIV with FDCs indicated that some virus may be retained on FDCs for years, where it can potentially reignite infection if treatment is interrupted. In that study, an approximate expression was used to estimate the time an individual virion remains on FDCs during therapy. Here, we determine the conditions under which this approximation is valid, and we develop expressions for the time a virion spends in any bound state and for the effect of rebinding on retention. We find that rebinding, which is influenced by diffusion, may play a major role in retention of HIV on FDCs. We also consider the possibility that HIV is retained on B cells during therapy, which like FDCs also interact with HIV. We find that virus associated with B cells is unlikely to persist during therapy.

Artificial in vivo antigen presentation by the APCs and subsequent T-cell activation: a feasibility analysis

Inappropriate antigen presentation by the antigen-presenting cells (APCs) is a cause of various diseases. One of the ways to combat these diseases is to immobilize the APCs near the infected tissue or a tissue which is susceptible to an antigen. The antigen is presented by the APCs present in the immobilized form on an implant and these upon binding to T(H)-cells result in triggering of a cascade of events as part of the natural immune response leading to the destruction of the antigen. This system has been modeled as a dialysis bag containing immobilized receptors inside the bag and the ligand diffusing out of the bag. The simulations show that by using the implant, the concentration of the ligand that has diffused into the tissue matrix can be substantially reduced and by suitably choosing the coupler size, the T(H)-cells can also effectively be activated.

T-cell activation by soluble MHC oligomers can be described by a two-parameter binding model

T-cell activation is essential for initiation and control of immune system function. T cells are activated by interaction of cell-surface antigen receptors with major histocompatibility complex (MHC) proteins on the surface of other cells. Studies using soluble oligomers of MHC-peptide complexes and other types of receptor cross-linking agents have supported an activation mechanism that involves T cell receptor clustering. Receptor clustering induced by incubation of T cells with MHC-peptide oligomers leads to the induction of T-cell activation processes, including downregulation of engaged receptors and upregulation of the cell-surface proteins CD69 and CD25. Dose-response curves for these T-cell activation markers are bell-shaped, with different maxima and midpoints, depending on the valency of the soluble oligomer used. In this study, we have analyzed the activation behavior using a mathematical model that describes the binding of multivalent ligands to cell-surface receptors. We show that a simple equilibrium binding model accurately describes the activation data for CD4(+) T cells treated with MHC-peptide oligomers of varying valency. The model can be used to predict activation and binding behavior for T cells and MHC oligomers with different properties.

Kinetic proofreading models for cell signaling predict ways to escape kinetic proofreading

In the context of cell signaling, kinetic proofreading was introduced to explain how cells can discriminate among ligands based on a kinetic parameter, the ligand-receptor dissociation rate constant. In the kinetic proofreading model of cell signaling, responses occur only when a bound receptor undergoes a complete series of modifications. If the ligand dissociates prematurely, the receptor returns to its basal state and signaling is frustrated. We extend the model to deal with systems where aggregation of receptors is essential to signal transduction, and present a version of the model for systems where signaling depends on an extrinsic kinase. We also investigate the kinetics of signaling molecules, "messengers," that are generated by aggregated receptors but do not remain associated with the receptor complex. We show that the extended model predicts modes of signaling that exhibit kinetic discrimination for some range of parameters but for other parameter values show little or no discrimination and thus escape kinetic proofreading. We compare model predictions with experimental data.

Steric effects on multivalent ligand-receptor binding: exclusion of ligand sites by bound cell surface receptors

Steric effects can influence the binding of a cell surface receptor to a multivalent ligand. To account for steric effects arising from the size of a receptor and from the spacing of binding sites on a ligand, we extend a standard mathematical model for ligand-receptor interactions by introducing a steric hindrance factor. This factor gives the fraction of unbound ligand sites that are accessible to receptors, and thus available for binding, as a function of ligand site occupancy. We derive expressions for the steric hindrance factor for various cases in which the receptor covers a compact region on the ligand surface and the ligand expresses sites that are distributed regularly or randomly in one or two dimensions. These expressions are relevant for ligands such as linear polymers, proteins, and viruses. We also present numerical algorithms that can be used to calculate steric hindrance factors for other cases. These theoretical results allow us to quantify the effects of steric hindrance on ligand-receptor kinetics and equilibria.